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In geometry, the rhombic dodecahedron is a convex polyhedron with 12 congruent rhombic faces. It has 24 edges, and 14 vertices of 2 types. As a Catalan solid, it is the dual polyhedron of the cuboctahedron. As a parallelohedron, the rhombic dodecahedron can be used to tesselate its copies in space creating a rhombic dodecahedral honeycomb.
Several nonconvex uniform polyhedra, including the tetrahemihexahedron, cubohemioctahedron, octahemioctahedron, small rhombihexahedron, small icosihemidodecahedron, and small dodecahemidodecahedron, have antiparallelograms as their vertex figures, the cross-sections formed by slicing the polyhedron by a plane that passes near a vertex, perpendicularly to the axis between the vertex and the center.
The boundary of a cross-section in three-dimensional space that is parallel to two of the axes, that is, parallel to the plane determined by these axes, is sometimes referred to as a contour line; for example, if a plane cuts through mountains of a raised-relief map parallel to the ground, the result is a contour line in two-dimensional space ...
A central cross section of a regular tetrahedron is a square. The two skew perpendicular opposite edges of a regular tetrahedron define a set of parallel planes. When one of these planes intersects the tetrahedron the resulting cross section is a rectangle. [11] When the intersecting plane is near one of the edges the rectangle is long and skinny.
The ten-of-diamonds can be dissected in an octagonal cross-section between the two rhombic faces. It is a decahedron with 12 vertices, 20 edges, and 10 faces (4 triangles, 4 trapezoids, 1 rhombus, and 1 isotoxal octagon). Michael Goldberg labels this polyhedron 10-XXV, the 25th in a list of space-filling decahedra. [2]
If the areas of the two parallel faces are A 1 and A 3, the cross-sectional area of the intersection of the prismatoid with a plane midway between the two parallel faces is A 2, and the height (the distance between the two parallel faces) is h, then the volume of the prismatoid is given by [3] = (+ +).
The icosidodecahedron is an Archimedean solid, meaning it is a highly symmetric and semi-regular polyhedron, and two or more different regular polygonal faces meet in a vertex. [5] The polygonal faces that meet for every vertex are two equilateral triangles and two regular pentagons, and the vertex figure of an icosidodecahedron is {{nowrap|(3 ...
A polyhedron that has a midsphere is said to be midscribed about this sphere. [1] When a polyhedron has a midsphere, one can form two perpendicular circle packings on the midsphere, one corresponding to the adjacencies between vertices of the polyhedron, and the other corresponding in the same way to its polar polyhedron, which has the same ...